Stationary phase method on supermanifolds The classical stationary phase method computes an asymptotic behavior of an integral $\int_M f(x)e^{-\frac{1}{h}S(x)} dx$ as $h\to +0$, where $M$ is (say, compact) manifold in terms of the critical points of the function $S$.

I am wondering if there exists a generalization of the stationary phase method to supermanifolds.

 A: The stationary phase approximation is strictly easier for fermionic manifolds than for bosonic ones. Indeed, suppose $M = \mathbb{R}^{0|n}$ is a purely odd supermanifold, with (odd) coordinates $x_1,\dots,x_n$. (Locally every supermanifold is $\mathbb{R}^{m|n} = \mathbb{R}^m \times \mathbb{R}^{0|n}$; you already understand stationary phase in the bosonic directions, so I'll just focus on the fermionic ones.) Up to rescaling, the unique translation-invariant volume form on $\mathbb{R}^{0|n}$ is
$$ \int f(x) \mathrm{d}^nx \propto f^{(n)}_{1\dots n} $$
where the Taylor expansion of $f$ is
$$ f(x) = f^{(0)} + f^{(1)}_i x_i + \frac{f^{(2)}_{ij}}2 x_i x_j + \dots + f^{(n)}_{1\dots n} x_1 \dots x_n.$$
The Taylor coefficient $f^{(k)}_{i_1\dots i_k}$ is totally antisymmetric in the indices, and of course I am summing over repeated indices. Since the $x_i$ are odd, they are nilpotent, and so the Taylor expansion is exact.
In particular, the fermionic Gaussian integral is
$$ \int e^{A_{ij} x_i x_j} \mathrm{d}^n x \propto \mathrm{Pf}(A), \qquad (*)$$
where $\mathrm{Pf}$ is the Pfaffian of the antisymmetric matrix $A$. It enjoys $\mathrm{Pf}(A)^2 = \mathrm{det}(A)$, and so this should remind you of the bosonic statement that for a positive-definite symmetric matrix $B$,
$$ \int e^{-B_{ij} y_i y_j} \mathrm{d}^n y \propto \frac{1}{\sqrt{\mathrm{det}B}}. \qquad (**)$$
Of course you may, if you choose, adjust some factors of $2\pi$.
The full asymptotics of the bosonic stationary phase approximation follow from (**) together with integration by parts (or, equivalently, turning on a chemical potential $J_i y_i$ in the integrand). The same is true for the fermionic integral: integration by parts works just as soon as your volume form is translation-invariant, and so you can combine (*) with integration by parts to produce a formula for $\int_{\mathbb{R}^{0|n}} f(x) e^{-S(x)}\mathrm{d}^n x$. The formula is exact, simply because all functions of a fermionic variable are polynomials.
